In [1]}, Wyllard made some explicit proposals for how the \(S\) -duality maps should act on rigid surface operators based on the invariants of partitions.
The invariant fingerprint based on Kazhdan-Lusztig map for the classical groups [2]}[3]}.
The Kazhdan-Lusztig map is a map from the unipotent conjugacy classes to the set of conjugacy classes of the Weyl group. It can be extended to the case of rigid semisimple conjugacy classes[4]}. The rigid semisimple conjugacy classes and the conjugacy classes of the Weyl group are described by pairs of partitions \((\lambda ^{^{\prime }};\lambda ^{^{\prime \prime }})\)  and pairs of partitions \([\alpha ;\beta ]\) , respectively. The fingerprint invariant is a map between these two classes of objects. There is another invariant symbol  related to the the Springer correspondence[5]}.
